Standard error
- Measure
The standard error measures how much the mean would vary if the study were repeated on another sample. It falls as the sample grows.
What it is
The standard error (SE) of the mean is the standard deviation of the means that would be obtained by repeating the sampling many times. It is the standard deviation of the data divided by the square root of n. While the standard deviation describes how much patients differ from each other, the standard error describes how precise the estimate is, and it is the basis of confidence intervals and hypothesis tests.
Formula and reading
- SE of a mean = SD / √n.
- SE of a proportion = √[p × (1 − p) / n].
- Reading: a small SE means a precise estimate; it says nothing about whether the estimate is biased.
Worked example
- Data: in 36 adults from one neighbourhood, systolic pressure has a mean of 128 mmHg and an SD of 18 mmHg.
- SE = 18 / √36 = 18 / 6 = 3 mmHg.
- 95% CI ≈ 128 ± 1.96 × 3 = 128 ± 5.9, that is 122.1 to 133.9 mmHg (rounded).
- If 144 adults were measured with the same SD: SE = 18 / √144 = 18 / 12 = 1.5 mmHg.
- Quadrupling n (36 to 144) halves the SE (3 to 1.5) and halves the width of the interval.
- The SD stays near 18 mmHg: adults do not become more alike because more of them are measured.
- Reading: a bigger sample gives a more precise mean, but it does not change how much people vary.
Markers
- SE
- SD
- √n
How to read it
- When reading a paper, check whether a chart’s error bars are SD, SE or 95% CI: they say different things.
- A mean with a tiny SE from a biased study is still wrong: precision does not fix bias.
Common pitfalls
- Using the SE to describe how variable the patients are (that is the SD’s job).
- Forgetting that the SE depends on n: two studies with the same SD can have very different SEs.
Test yourself
Think of the answer, then tap the question to see it.
How is the standard error of a mean calculated?
Divide the standard deviation by the square root of the sample size: SE = SD / √n.
How much must the sample grow to halve the standard error?
Fourfold, because the SE depends on the square root of n.
What is the difference between the standard deviation and the standard error?
The SD measures the spread of individual data points; the SE measures the precision of the mean estimated from them.
With SD = 20 and n = 100, what is the standard error?
SE = 20 / √100 = 20 / 10 = 2.
Connections
Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.
Sources: Principles of Epidemiology | Lesson 2 - Section 7 (Centers for Disease Control and Prevention, 2012); Confidence Limits for the Mean (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012).
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Representative model, for educational purposes only.